98 Formule pentru algebra disponibile

$\frac{a}{c} = \frac{b}{d}$

$\|\vec{u}\| = \sqrt{a^2 + b^2}$

$\|\vec{v}\| = \sqrt{x^2 + y^2 + z^2}$

$\vec{v_1} \cdot \vec{v_2} = x_1x_2 + y_1y_2 + z_1z_2$

$\log_a x = y \iff a^y = x$

$a^{\log_a x} = x$

$\log_a x^k = k \cdot \log_a x$

$\log_a (xy) = \log_a x + \log_a y$

$\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

$a_{n+1} = a_n + r$

$a_n = a_1 + (n-1)r$

$S_n = \frac{(a_1 + a_n) \cdot n}{2}$

$b_n = b_1 \cdot q^{n-1}$

$S_n = \frac{b_1(q^n - 1)}{q - 1}, q \neq 1$

$f = a_nX^n + a_{n-1}X^{n-1} + ... + a_1X + a_0$

$grad(f + g) = grad\ f + grad\ g$

$f = gq + r$

$f(a) = 0 \iff (X - a) | f$

$f = a_n (X-x_1)^{\alpha_1} (X-x_2)^{\alpha_2} ...(X-x_k)^{\alpha_k}$

$x_1 + x_2 + ... + x_n = -\frac{a_{n-1}}{a_n}$

$x_1x_2...x_n = (-1)^n\frac{a_0}{a_n}$

$z = a + bi$

$\overline{z} = a - bi$

$|z| = \sqrt{a^2 + b^2}$

$|z_1 + z_2| \leq |z_1| + |z_2|$

$P_n = n!$

$A_n^k = \frac{n!}{(n-k)!}$

$C_n^k = \frac{n!}{k!(n-k)!}$

$C_n^k = C_n^{n-k}$

$C_n^k = C_{n-1}^k + C_{n-1}^{k-1}$

$(a+b)^n = \sum_{k=0}^n C_n^k \cdot a^{n-k} \cdot b^k$

$T_{k+1} = C_n^k \cdot a^{n-k} \cdot b^k$

$\sum_{k=0}^n C_n^k = 2^n$

$\frac{T_{k+1}}{T_k} = \frac{n-k+1}{k} \cdot \frac{b}{a}$

$\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma(1) & \sigma(2) & \cdots & \sigma(n) \end{pmatrix}$

$\varepsilon(\sigma) = (-1)^{m(\sigma)}$

$\varepsilon(\sigma_1 \circ \sigma_2) = \varepsilon(\sigma_1) \cdot \varepsilon(\sigma_2)$

$A_n^k = \frac{n!}{(n-k)!}$

$A_n^n = n!$

$A_n^k = n \cdot A_{n-1}^{k-1}$

$A_n^k = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot (n-k+1)$

$A_n^k = C_n^k \cdot k!$

$\sum_{k=0}^n A_n^k = \sum_{k=0}^n \frac{n!}{(n-k)!} = n! \cdot e - \left\lfloor n! \cdot e \right\rfloor$

$ax^2 + bx + c = 0, \quad a \neq 0, \quad a,b,c \in \mathbb{R}$

$f(x) = ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{\Delta}{4a}$

$\Delta = b^2 - 4ac$

$x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}$

$S = x_1 + x_2 = -\frac{b}{a}, \quad P = x_1 \cdot x_2 = \frac{c}{a}$

$f(x) = ax^2 + bx + c, a\neq 0$

$V\left(-\frac{b}{2a}, -\frac{\Delta}{4a}\right)$

$x = -\frac{b}{2a}$

$Z = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

$|a| = \begin{cases} a, & \text{dacă } a \geq 0 \\ 0, & \text{dacă } a = 0 \\ -a, & \text{dacă } a < 0 \end{cases}$

$|a \cdot b| = |a| \cdot |b|$

$\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$

$|a| - |b| \leq |a + b| \leq |a| + |b|$

$a^n = \underbrace{a \cdot a \cdot a \cdot ... \cdot a}_{\text{de n ori}}$

$a^m \cdot a^n = a^{m+n}$

$(a^m)^n = a^{m \cdot n}$

$\emptyset$

$a \in E$

$a \notin E$

$A \subset B$

$A \cup B = \{x | x \in A \text{ sau } x \in B\}$

$A \cap B = \{x | x \in A \text{ și } x \in B\}$

$A \cap B = \emptyset$

$A - B = \{x | x \in A \text{ și } x \notin B\}$

$*: M \times M \to M$

$(\forall) x, y \in H \Rightarrow x * y \in H$

$(x * y) * z = x * (y * z)$

$x * y = y * x$

$x * e = e * x = x$

$x * x^{-1} = x^{-1} * x = e$

$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$

$\det(A) = \sum_{\sigma \in S_n} \varepsilon(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}...a_{n\sigma(n)}$

$\det(A) = a_{11}a_{22} - a_{12}a_{21}$

$\det(AB) = \det(A) \cdot \det(B)$

$A \cdot A^{-1} = A^{-1} \cdot A = I_n$

$A^{-1} = \frac{1}{\det(A)} \cdot A^*$

$f : A \to B, \forall x, y \in A, x \neq y \Rightarrow f(x) \neq f(y)$

$f : A \to B, \forall y \in B, \exists x \in A : f(x) = y$

$f : A \to B \text{ este bijectivă } \Leftrightarrow f \text{ este injectivă și surjectivă}$

$f : A \to B \text{ este inversabilă } \Leftrightarrow \exists g : B \to A, f \circ g = 1_B \text{ și } g \circ f = 1_A$

$f : I \to \mathbb{R}, \forall x, y \in I, \forall a, b \geq 0, a + b = 1 : f(ax + by) \leq af(x) + bf(y)$

$f : I \to \mathbb{R}, \forall x, y \in I, \forall a, b \geq 0, a + b = 1 : f(ax + by) \geq af(x) + bf(y)$

$3x^2y, -5ab^3, 2xyz$

$3x^2y^3z$

$grad(3x^2y^3z) = 2 + 3 + 1 = 6$

$(3x^2y) \cdot (2xy^3) = 6x^3y^4$

$(3x^2y)^3 = 27x^6y^3$

$\frac{6x^3y^2}{2xy} = 3x^2y$

$f: A \rightarrow B$

$f: A \rightarrow B, A = \{1, 2, 3\}, B = \{a, b, c\}, f(1) = a, f(2) = b, f(3) = c$

$f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = 2x + 1$

$\{(x, y) | x \in A, y = f(x)\}$

$f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = ax + b$

$G_f \cap Oy = A(0, b)$

$G_f \cap Ox = B(-\frac{b}{a}, 0)$